This is the first of three sessions by Dr Joel Feinstein on how and why we do proofs.

Dr Feinstein's blog is available at http://explainingmaths.wordpress.com/

The aim of this session is to motivate students to understand why we might want to do proofs, why proofs are important, and how they can help us. In particular, the student will learn the following: proofs can help you to really see WHY a result is true; problems that are easy to state can be hard to solve (Fermat's Last Theorem); sometimes statements which appear to be intuitively obvious may turn out to be false (Simpson's paradox); the answer to a question will often depend crucially on the definitions you are working with.

Target audience: suitable for anyone with a knowledge of elementary algebra and prime numbers, as may be

obtained by studying A level mathematics.

This popular maths talk by Dr Joel Feinstein gives an introduction to various different kinds of infinity, both countable and uncountable. These concepts are illustrated in a somewhat informal way using the notion of Hilbert's infinite hotel. In this talk, the hotel manager tries to fit various infinite collections of guests into the hotel. The students should learn that many apparently different types of infinity are really the same size. However, there are genuinely "more" real numbers than there are positive integers, as is shown in the more challenging final section, using Cantor's diagonalization argument.

This last part of the talk is relatively technical, and is probably best suited to first-year mathematics undergraduates, or advanced maths A level students. Others may find the technical details hard to follow, and should focus on the overview.

Dr Joel Feinstein's blog is available at http://explainingmaths.wordpress.com/

This is the first of two sessions on how to do proofs. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/

The aim of these sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. In particular, we focus on how to start proofs, and how and when to use definitions and known results. With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more creative and interesting aspects of constructing proofs.

Part I is suitable for anyone with a knowledge of elementary algebra (including odd numbers, multiples of eight and the binomial theorem for expanding powers of (a+b)), and functions from

the set of real numbers to itself (odd functions, even functions, multiplication and composition of functions).

In this workshop, Dr Feinstein helps students to understand convergence of sequences in finite-dimensional Euclidean space using his terminology of sets absorbing sequences (absorption).

For more on the advantages of this terminology, see Dr Feinstein's blog post at http//wp.me/posHB-c

See also Dr Feinstein's blog, Explaining Mathematics,

http//explainingmaths.wordpress.com/

In this workshop, Dr Feinstein helps students to understand convergence of sequences in finite-dimensional Euclidean space using his terminology of sets absorbing sequences (absorption).

For more on the advantages of this terminology, see Dr Feinstein's blog post at http//wp.me/posHB-c

See also Dr Feinstein's blog, Explaining Mathematics,

http//explainingmaths.wordpress.com/

This is the first of two sessions on how to do proofs. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/

The aim of these sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. In particular, we focus on how to start proofs, and how and when to use definitions and known results. With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more creative and interesting aspects of constructing proofs.

Part I is suitable for anyone with a knowledge of elementary algebra (including odd numbers, multiples of eight and the binomial theorem for expanding powers of (a+b)), and functions from

the set of real numbers to itself (odd functions, even functions, multiplication and composition of functions).

This is the first of three sessions by Dr Joel Feinstein on how and why we do proofs.

Dr Feinstein's blog is available at http://explainingmaths.wordpress.com/

The aim of this session is to motivate students to understand why we might want to do proofs, why proofs are important, and how they can help us. In particular, the student will learn the following: proofs can help you to really see WHY a result is true; problems that are easy to state can be hard to solve (Fermat's Last Theorem); sometimes statements which appear to be intuitively obvious may turn out to be false (Simpson's paradox); the answer to a question will often depend crucially on the definitions you are working with.

Target audience: suitable for anyone with a knowledge of elementary algebra and prime numbers, as may be

obtained by studying A level mathematics.

This popular maths talk by Dr Joel Feinstein gives an introduction to various different kinds of infinity, both countable and uncountable. These concepts are illustrated in a somewhat informal way using the notion of Hilbert's infinite hotel. In this talk, the hotel manager tries to fit various infinite collections of guests into the hotel. The students should learn that many apparently different types of infinity are really the same size. However, there are genuinely "more" real numbers than there are positive integers, as is shown in the more challenging final section, using Cantor's diagonalization argument.

This last part of the talk is relatively technical, and is probably best suited to first-year mathematics undergraduates, or advanced maths A level students. Others may find the technical details hard to follow, and should focus on the overview.

Dr Joel Feinstein's blog is available at http://explainingmaths.wordpress.com/